A good way to start a hard math problem is by playing around with it. Poking at it. Trying things to see what does and doesn’t work and to figure out why that is.
For a certain type of math problems, it’s helpful to just guess and then analyze why that guess isn’t right.
I don’t know if you’ve ever played the game Mastermind, but Mastermind is exactly this idea. One player hides 4 pin colors, and the second player has to guess what the colors are and where they’re placed. Each turn player two is given information on how much ze got wrong and how wrong it was. The only way to start is with a completely blind guess. If you guess right on the first try, the game isn’t very much fun. That means you won by luck and not by being able to actually play the game.
DC1 had never heard of such a thing before we got the Hard Math book. Ze was completely and totally frustrated by the first challenge problem (What is the largest possible answer to 782 + ABC =? [with carrying 1s above and above/left of the 7]?) because ze thought ze should just be able to do a math problem. Even hard math problems were hard because ze was prone to make mistakes, and all one had to do was not make mistakes. This idea that you have to learn about the problem first and maybe try a few things out was completely foreign to hir.
Life is like that too. You can plan and plot and analyze situations, but sometimes that takes more time (and provides less information) than just doing and seeing what happens. Sometimes you get what you want on the first try, but more often, you get clear information on what you need to do better and how and why.
Sometimes you have to fail before succeeding, and it’s the failure(s) itself that is instrumental to your eventual success.